However, there are some complicated matrix multiplications in the last three lines of formula (REF ) at the top of this page arising from \(\mathbf {P}_l, \forall l\) , which make it difficult to identify the variational pdf \(Q^\dagger (\mathbf {G})\) . To reveal its structure, we utilize the property of the vectorization operation that \(\mathrm {Tr}(\mathbf {A}\mathbf {X}\mathbf {B}\mathbf {X}^T)=\mathrm {vec}(\mathbf {X})^T(\mathbf {B}\otimes \mathbf {A}^T)\mathrm {vec}(\mathbf {X})\)  with \(\mathbf {X}\in \mathbb {C}^{m\times n}\) , \(\mathbf {A}\in \mathbb {C}^{m\times m}\)  and \(\mathbf {B}\in \mathbb {C}^{n\times n}\)  [1]}, and then the terms inside the second \(\exp (\cdot )\)  in (REF ) can be reformulated as
\(& -\mathrm {Tr}\left(\sum _{l=1}^{L}\mathbf {P}_l^H\mathbf {P}_l\mathbf {G}\Xi _l^{-1}\mathbf {G}^H\right.\left.-\mathbf {P}_l^H\mathbf {P}_l\Phi _l \Xi _l ^{-1}\mathbf {G}^H\right.\nonumber \\&\left.-\mathbf {P}_l^H\mathbf {P}_l\mathbf {G}\Xi _l^{-1}\Phi _l^H\right)\nonumber \\& = -\mathrm {Tr}\left(\sum _{l=1}^{L}\Xi _l^{-1}\mathbf {G}^H\mathbf {P}_l^H\mathbf {P}_l\mathbf {G}\right.\nonumber \\&\left.-\Xi _l ^{-1}\Phi _l^H \mathbf {P}_l^H\mathbf {P}_l \mathbf {G}-\Xi _l^{-1}\mathbf {G}^H\mathbf {P}_l^H\mathbf {P}_l\Phi _l\right)\nonumber \\&=\sum _{l=1}^{L}-\mathrm {vec}(\mathbf {G}^H)^H\left(\mathbf {P}_l^H\mathbf {P}_l\otimes \Xi _l^{-1}\right)\mathrm {vec}(\mathbf {G}^H)\nonumber \\&+\mathrm {vec}(\Phi _l^H)^H \left(\mathbf {P}_l^H\mathbf {P}_l\otimes \Xi _l^{-1}\right)\mathrm {vec}(\mathbf {G}^H) \nonumber \\&+\mathrm {vec}(\mathbf {G}^H)^H\left(\mathbf {P}_l^H\mathbf {P}_l\otimes \Xi _l^{-1}\right)\mathrm {vec}(\Phi _l^H).\) 
